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Hydrodynamics of a one-dimensional granular medium

Phys. Fluids 7, 507 (1995); doi:10.1063/1.868648

Issue Date: March 1995

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N. Sela and I. Goldhirsch
Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel
The question whether one-dimensional granular systems can be described by hydrodynamic equations is the main theme of the present work. Numerical simulations are used to create a database with which theory is compared. The system investigated in the numerical work is that of a one-dimensional collection of point particles colliding inelastically. The dependence of the dynamical properties on both the degree of inelasticity and the number of particles is investigated. A hydrodynamic theory which describes the large-scale motion of such systems has been developed. It is shown that the standard set of hydrodynamic fields (density, velocity, and granular temperature) is insufficient for this purpose and that an additional hydrodynamic field corresponding to the third moment of the fluctuating velocity field must be added to that set. The results of a linear stability analysis of the derived hydrodynamic equations are in a close agreement with those of the numerical simulations. The question of the effects of velocity correlations on the hydrodynamics is addressed as well. It is shown that these correlations, though not negligible, do not affect the hydrodynamic equations. The form of the single particle initial distribution function is shown to slightly affect the form of the hydrodynamic equations for transient times. Except for this minor effect the hydrodynamic equations possess a universal form. Possible implications for higher dimensional systems are mentioned. ©1995 American Institute of Physics.
History: Received 27 July 1994; accepted 15 November 1994
Permalink: http://link.aip.org/link/?PHFLE6/7/507/1
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KEYWORDS and PACS

Keywords
PACS
  • 05.20.Dd
    Statistical physics and thermodynamics Statistical mechanics Kinetic theory
  • 46.10.+z
    Classical mechanics Mechanics of discrete systems
  • 51.10.+y
    Physics of gases Kinetic and transport theory of gases
  • 82.70.-y
    Physical chemistry Disperse systems
  • YEAR: 1995

PUBLICATION DATA

ISSN:
1070-6631 (print)   1089-7666 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (23)
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  2. M. Babic, “On the stability of rapid granular flows,” J. Fluid Mech. 254, 127 (1993).
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  4. I. Goldhirsch and M. L. Tan, “Cluster formation in granular shear flow" (preprint, 1994).
  5. I. Goldhirsch, M. L. Tan, and G. Zanetti, “A molecular dynamical study of granular fluids: the unforced granular gas in two dimensions,” J. Sci. Comput. 8, 1 (1993).
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  11. J. T. Jenkins and M. W. Richman, “Plane simple shear of smooth inelastic circular disks: The anisotropy of the second moment in the dilute and dense limits,” J. Fluid Mech. 192, 313 (1988).
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  13. P. K. Haff, “Grain flow as a fluid mechanical phenomenon,” J. Fluid Mech. 134, 401 (1983).
  14. In this paper (except in Appendix B) we consider f as the distribution function in the velocity space alone, i.e., the commonly used f normalized by the density rho.
  15. C. K. K. Lun, S. B. Savage, D. J. Jeffrey, and N. Chepumiy, “Kinetic theories for granular flows: Inelastic particles in Couette flow and slightly inelastic particles in general flow field,” J. Fluid Mech. 140, 223 (1984).
  16. J. T. Jenkins and M. W. Richman, “Kinetic theory for plane flows of a dense gas of identical, rough, inelastic circular disks,” Phys. Fluids 28, 3485 (1985).
  17. H. Grad, “On the kinetic theory of rarefied gases,” Comm. Pure Appl. Math. 2, 331 (1949).
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  19. The same result can be obtained directly from the “test particle equation”10 by multiplying it by upsilon2 and integrating over upsilon. This procedure results in an equation of motion for the granular temperature with an energy sink term identical to the one given in Eq. (36).
  20. S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases, 3rd ed. (Cambridge University Press, Cambridge, 1970).
  21. The regression slope is based on all points in the case of e = 0.995 and on the first four points in the case of = 0.99, i.e., before the “inelastic collapse” sets in.
  22. Notice that if we had taken sigma =zeta in the three-particle distribution function then the average given in Eq. (80) would have vanished.
  23. The reason that the simulated critical value of e is somewhat lower than the one predicted by Eq. (62), may be attributed to the fact that the Gaussian distribution is not a solution of the homogeneous Boltzmann equation. The latter should evolve to a two-hump distribution (cf. Fig. 3). As the instability (discussed in this paper) evolves so does the distribution function. Considering the fact that an approximate flat distribution is a transient stage in the development of the distribution function from its Gaussian shape to two-hump form, it is clear that the exact critical value of e should be between that predicted by using the Gaussian distribution and the one predicted by employing the flat distribution.

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